How do I extrude this cylinder?
1) Keeping modifiers unapplied

Modeling without applying modifiers can be quite tricky.

Tring to keep the Solidify modifier, which basically extrude faces along their normals, isn't probably the best choiche for your task because your reference object has a different depth extrusion on one side.



You could have done that by stacking two Solidify modifiers, but unfortunately currently faces created by another modifier cannot be pickedlike in other software packages (3dsMax...). Probably, with a bit of work the closest thing you can get is something like this:



where I had to Inset faces and pick only a few of them manually in order to put them in a Vertex group which would allowed me to set a different Factor in the modifier for the object's faces.



From the Blender manual page of the Solidify modifier;


  Vertex Group
  
  Only vertices in this group are solidified - their weights are multiplied by the thickness, so vertices with lower weights will be
  less thick.


and


  Factor
  
  How much the vertex weights are taken into account.
  
  
  On 0.0, vertices with zero weight will have no thickness at all.
  On 0.5, vertices with zero weight will be half as thick as those with full weight.
  On 1.0, the weights are ignored and the thickness value is used for every vertex.
  




To make the edges sligtly thinner toward the center you can work (for example) on the inner edges of the Full Weight faces:





2) Applying modifiers

If is suitable for the project, it's probably easier to apply the Solidifiy modifier it seems you are using (or extrude the faces inward as in the picture below)



then use the Checker deselect tool to pick one face every two and "Extrude region (vertex normals)" to build the gear tooth.



Pick the iner faces, set Individual origin Pivot, Normal orientation and then scale each faces along the Normal-X axis by pressing in sequence SXX followed by the wanted value.